1 Answer
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The right-hand column shows the positive integers less than 21 (the modulus) that have quadratic residue equal to the values in the left-hand column. So, for example, the integers 1, 8, 13 and 20 all have quadratic residue equal to 1 modulo 21. This means that their squares are congruent to 1 modulo 21. For example,

8 * 8 = 64 = 63 + 1 = 21 * 3 + 1 =. 0 + 1 mod 21 =. 1 mod 21

where I am using =. to represent congruency modulo 21. Similarly,

13 * 13 = 169 = 168 + 1 = 21 * 8 + 1 =. 0 + 1 mod 21 =. 1 mod 21

and

20 * 20 = 400 = 399 + 1 = 21 * 19 + 1 =. 0 + 1 mod 21 =. 1 mod 21.

Finding these numbers is called finding square roots mod n. You can find them using the Chinese Remainder Theorem (assuming that you can factor the modulus).